Parabolic Besov Regularity for the Heat Equation

Abstract

We obtain parabolic Besov smoothness improvement for temperatures on cylindrical regions based on Lipschitz domains. The results extend those for harmonic functions obtained by S. Dahlke and R. DeVore using the wavelet description of Besov regularity.

On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

“…Summing over all indices (i, j, k) ∈ Λ 0 j and applying Hölder's inequality with exponents p/τ > 1 and p/(p − τ ) one finds Obviously, the sums on the right hand side converge if, and only if, α ∈ 0, γ ∧ ν d θ := 2θ ′ (p − 1)/(p − 2) − dp/(p − 2), where θ/p + θ ′ /p ′ = d with 1/p + 1/p ′ = 1. Moreover, by Lemma 2.1(i) we can choose a sequenceφ n ⊆ C ∞ 0 (G (1) ) approximating ϕ • φ in H 1 p,θ−p (G (1) ). Furthermore, a consequence of our assumptions is, that for all n ∈ N, with probability one, the equality u(t, ·),φ n = u(0, ·),φ n + t 0 f (s, ·),φ n ds + Hence, since H 1 p,θ−p (G (1) ) is continuously embedded in H 1 p ′ ,θ ′ −p ′ (G (1) ) ∩ L p ′ ,θ ′ (G (1) ) ∩ Lp ,θ (G (1) ), the assertion follows for the case p > 2.…”

“…Moreover, by Lemma 2.1(i) we can choose a sequenceφ n ⊆ C ∞ 0 (G (1) ) approximating ϕ • φ in H 1 p,θ−p (G (1) ). Furthermore, a consequence of our assumptions is, that for all n ∈ N, with probability one, the equality u(t, ·),φ n = u(0, ·),φ n + t 0 f (s, ·),φ n ds + Hence, since H 1 p,θ−p (G (1) ) is continuously embedded in H 1 p ′ ,θ ′ −p ′ (G (1) ) ∩ L p ′ ,θ ′ (G (1) ) ∩ Lp ,θ (G (1) ), the assertion follows for the case p > 2. The same arguments can be used to prove the case p = 2: Just replacep by 2, andθ by θ ′ = 2d − θ and use the estimate and…”

“…Using the a-priori estimates from Section 4, the right hand side of the above inequality can be estimated by suitable norms of f and g if u is the solution to the corresponding SPDE (Theorem 7.5). Let us mention the related work [1] on the Besov regularity for the deterministic heat equation. The authors study the regularity of temperatures in terms of anisotropic Besov spaces of type B α/2,α τ,τ ((0, T ) × O), 1/τ = α/d + 1/p.…”

Improvement of Besov Regularity for Solutions of the Fractional Laplacian

“…It is worth noticing that in contrast to the local cases associated with the harmonic functions in [3] and the temperatures in [1], now the B λ p regularity is required on the whole space R n and that the improvement is only in D.…”

“…In order to obtain improved results for the Besov regularity of solutions of (− ) s f = 0 in the spirit of [3] and [1], our formula seems to be more suitable because we can get explicit estimates for the gradients of the mean value kernel. Regarding Besov regularity of harmonic functions, see also [8].…”